Question

State whether the following statement is true or false. Give reasons for your answers. Square numbers can be written as the sum of two odd numbers.

Answer 1

Hint: Let us take some examples for different cases like if the square number is even or the square number is odd because if both cases are true then the statement will be true.

Complete step-by-step solution:
As we know that the odd numbers are those which are not directly divisible by 2.
Like 1, 3, 5, 7 etc are the odd numbers.
And even numbers are those which are directly divisible by 2.
Like 2, 4, 6, 8 etc are the even numbers.
So, now let us find whether the given statement is true or false for all different cases. If the statement is false for any of the cases, then the statement will be false otherwise it will be true.
Case 1:- If the square number is even.
Example: 4, 64 etc
4 = 1 + 3 and 64 = 1 + 63
So, if the square is even then the number can be written as the sum of two odd numbers.
Case 2:- If the square number is odd.
Example: 9, 25 etc
9 = 7 + 2 and 25 = 23 + 2
So, if the square number is odd then the number cannot be written as the sum of two odd numbers. It can be written as the sum of one odd and one even number.
Hence, the given statement is false because if the square number is odd then it cannot be written as the sum of two odd numbers.

Note: Square of odd number is always odd because if n is any natural number that 2n + 1 will be odd and square of 2n + 1 will be (2n + 1)*(2n + 1) = \[4{{\text{n}}^2}\] + 4n + 1, which is odd and square of even number 2n is 2n*2n = \[4{{\text{n}}^2}\], which is even. So, we can say that if the number is even then its square can be written as the sum of two odd numbers and if the number is odd its square can be written as the sum of one odd and one even number.